This time we are interested in solving the inhomogeneous wave
equation (IWE)

(11.52)

(for example) directly, without doing the Fourier transform(s) we did to
convert it into an IHE.

Proceeding as before, we seek a Green's function that satisfies:

(11.53)

The primary differences between this and the previous cases are a) the
PDE is hyperbolic, not elliptical, if you have any clue as to what that
means; b) it is now four dimensional - the ``point source'' is one that
exists only at a single point in space for a single instant in time.

Of course this mathematical description leaves us with a bit of an
existential dilemna, as physicists. We generally have little trouble
with the idea of gradually restricting the support of a distribution to
a single point in space by a limiting process. We just squeeze it down,
mentally. However, in a supposedly conservative Universe, it is hard
for us to imagine one of those squeezed down distributions of charge
just ``popping into existence'' and then popping right out. We can't
even do it via a limiting process, as it is a bit bothersome to
create/destroy charge out of nothingness even gradually! We are left
with the uncomfortable feeling that this particular definition is
nonphysical in that it can describe no actual physical sources -
it is by far the most ``mathematical'' or ``formal'' of the constructs
we must use. It also leaves us with something to understand.

One way we can proceed is to view the Green's functions for the IHE as
being the Fourier transform of the desired Green's function here!
That is, we can exploit the fact that:

(11.54)

to create a Fourier transform of the PDE for the Green's function:

(11.55)

(where I'm indicating the explicit dependence for the moment).

From the previous section we already know these solutions:

(11.56)

(11.57)

(11.58)

At this point in time 11.3 the only thing left
to do is to Fourier transform back - to this point in time:

(11.59)

(11.60)

(11.61)

(11.62)

so that:

(11.63)

(11.64)

Note that when we set , we basically asserted that
the solution is being defined without dispersion! If there is
dispersion, the Fourier transform will no longer neatly line up and
yield a delta function, because the different Fourier components will
not travel at the same speed. In that case one might still expect a
peaked distribution, but not an infinitely sharp peaked
distribution.

The first pair are generally rearranged (using the symmetry of the delta
function) and presented as:

(11.65)

and are called the retarded (+) and advanced (-) Green's
functions for the wave equation.

The second form is a very interesting beast. It is obviously a Green's
function by construction, but it is a symmetric combination of advanced
and retarded. Its use ``means'' that a field at any given point in
space-time
consists of two pieces - one half of it is due to
all the sources in space in the past such that the fields they emit are
contracting precisely to the point
at the instant and the
other half is due to all of those same sources in space in the
future such that the fields currently emerging from the point
at precisely arrive at them. According to this view, the field at
all points in space-time is as much due to the charges in the future as
it is those same charges in the past.

Again it is worthwhile to note that any actual field configuration
(solution to the wave equation) can be constructed from any of
these Green's functions augmented by the addition of an arbitrary
bilinear solution to the homogeneous wave equation (HWE) in primed and
unprimed coordinates. We usually select the retarded Green's function
as the ``causal'' one to simplify the way we think of an evaluate
solutions as ``initial value problems'', not because they are any more
or less causal than the others. Cause may precede effect in human
perception, but as far as the equations of classical electrodynamics are
concerned the concept of ``cause'' is better expressed as one of
interaction via a suitable propagator (Green's function) that may well
be time-symmetric or advanced.

A final note before moving on is that there are simply lovely papers
(that we hope to have time to study) by Dirac and by Wheeler and Feynman
that examine radiation reaction and the radiation field as constructed
by advanced and retarded Green's functions in considerable detail.
Dirac showed that the difference between the advanced and retarded
Green's functions at the position of a charge was an important
quantity, related to the change it made in the field presumably
created by all the other charges in the Universe at that point in
space and time. We have a lot to study here, in other words.

Using (say) the usual retarded Green's function, we could as usual write
an integral equation for the solution to the general IWE above for e.g.
:

(11.66)

where solves the HWE. This (with ) is essentially
equation (9.2), which is why I have reviewed this. Obviously we also
have

(11.67)

for
(the minus signs are in the differential equations
with the sources, note). You should formally verify that these
solutions ``work'' given the definition of the Green's function above
and the ability to reverse the order of differentiation and integration
(bringing the differential operators, applied from the left, in
underneath the integral sign).

Jackson proceeds from these equations by fourier transforming back into
a representation (eliminating time) and expanding the result to
get to multipolar radiation at any given frequency. However, because of
the way we proceeded above, we don't have to do this. We could
just as easily start by working with the IHE instead of the IWE and use
our HE Green's functions. Indeed, that's the plan, Stan...